Overview
The formula for the estimated values of {% y %} is given by
{% \hat{y} = \textbf{X} \vec{\beta} %}
This can be decomposed as
{% \textbf{X} \vec{\beta} = \sum_i \beta_i \vec{X}_i %}
(see matrix product)
That is, the estimated value of {% \vec{y} %} is the projection of {% \vec{y} %} onto the column space of {% \textbf{X} %}.